-
Analyzing Multistationarity in Chemical ReactionNetworks using
the Determinant Optimization Method
Bryan Félixa, Anne Shiub, Zev Woodstockc,
aDepartment of Mathematics, University of Texas at Austin, RLM
8.100, 2515 SpeedwayStop C1200, Austin, Texas 78712-1202, USA
bDepartment of Mathematics, Texas A&M University, Mailstop
3368, College Station,Texas 77843–3368, USA
cDepartment of Mathematics and Statistics, James Madison
University, Roop Hall 305,MSC 1911, Harrisonburg, Virginia 22807,
USA
Abstract
Multistationary chemical reaction networks are of interest to
scientists and
mathematicians alike. While some criteria for multistationarity
exist, obtain-
ing explicit reaction rates and steady states that exhibit
multistationarity for a
given network—in order to check nondegeneracy or determine
stability of the
steady states, for instance—is nontrivial. Nonetheless, we
accomplish this task
for a certain family of sequestration networks. Additionally,
our results allow
us to prove the existence of nondegenerate steady states for
some of these se-
questration networks, thereby resolving a subcase of a
conjecture of Joshi and
Shiu. Our work relies on the determinant optimization method,
developed by
Craciun and Feinberg, for asserting that certain networks are
multistationary.
More precisely, we implement the construction of reaction rates
and multiple
steady states which appears in the proofs that underlie their
method. Further-
more, we describe in detail the steps of this construction so
that other researchers
can more easily obtain, as we did, multistationary rates and
steady states.
Keywords: Mass-action kinetics, chemical reaction networks,
multistationarity, determinant optimization method, steady
states, degeneracy
1. Introduction
Although many dynamical systems arising in applications exhibit
bistability,
there is no complete characterization of such systems. Even for
the subclass
∗Please send correspondence to ZW.Email addresses:
[email protected] (Bryan Félix), [email protected]
(Anne Shiu), [email protected] (Zev Woodstock)
Preprint submitted to Applied Mathematics and Computation April
21, 2016
-
of chemical kinetics systems and even under the assumption of
mass-action
kinetics, which is the focus of this work, the problem is
difficult.5
Here we consider the simpler, yet still challenging, question:
which chemical
reaction networks are multistationary, i.e. which have the
capacity to exhibit
two or more steady-state concentrations with the same reaction
rates? Mathe-
matically, this asks: among certain parametrized families of
polynomial systems,
which admit multiple positive roots? Therefore, this is a real
algebraic geometry10
problem, and we do not expect an easy answer in general.
The first partial answers to this question are due to Feinberg,
Horn, and
Jackson in the 1970s. Their results in chemical reaction network
theory [1, 2]
(specifically, deficiency theory [3]) can preclude or guarantee
multistationarity
for certain classes of networks. For a survey of these and other
methods, see [4].15
Our work pertains to two related results: (1) a method for
“lifting” multiple
steady states from small networks to larger ones, and (2) the
so-called determi-
nant optimization method for certifying that a given network is
multistationary.
The lifting result, stated informally, is as follows: if a
chemical reaction
network contains an “embedded” network that is multistationary,
then the entire20
reaction network also is multistationary under certain
hypotheses [5]. Therefore
we are interested in cataloguing the multistationary networks
which contain no
embedded multistationary networks, because all larger
multistationary networks
contain at least one embedded multistationary subnetwork from
the catalogue.
As a step toward such a catalogue, Joshi and Shiu identified a
certain infinite25
family of chemical reaction networks K̃m,n to be of particular
interest among all
networks that include inflow and outflow reactions [4]. This
family is minimal,
in that it has no embedded subnetworks (with inflow and outflow
reactions)
that exhibit multistationarity. To analyze these networks, Joshi
and Shiu used
the second method for analyzing multistationarity mentioned
above.30
Developed by Craciun and Feinberg, the determinant optimization
method
can assert that a network is multistationary [6, 7]; as such, it
is a partial converse
to their results on “injective” reaction networks which
guarantee that a network
is not multistationary. This topic of injectivity has seen much
interest in recent
years (see [6, 8, 9] and the references therein); however, the
determinant opti-35
mization method has garnered comparatively little attention. The
only related
results that we are aware of are due to Banaji and Pantea [8],
Feliu [10], and
Müller et al. [9].
Using the determinant optimization method, Joshi and Shiu proved
that
K̃m,n is multistationary for all integers m ≥ 2 and odd integers
n ≥ 3 [4].40Furthermore, they conjectured that these networks can
exhibit multiple nonde-
2
-
generate steady states. (It is not guaranteed that the
determinant optimization
method produces nondegenerate steady states; we show this for
the first time
in Remark 3.11.) The significance of the conjecture is that if
it is true, then
K̃2,n would be the first example of an infinite family of
chemical reaction net-45
works with inflow and outflow reactions and at-most-bimolecular
reactants and
products—that is, minimal with respect to the embedding relation
among all
such networks which have the capacity to exhibit multiple
nondegenerate steady
states. Nondegeneracy is important because results that “lift”
multiple steady
states from embedded subnetworks or other typically smaller
networks require50
the steady states to be nondegenerate; a summary of such results
appears in [4,
§4]. Also, because trimolecular reactants/products are rather
uncommon inchemistry and K̃2,n is at most bimolecular, this family
of networks is of partic-
ular interest in chemical applications.
In the current work, we resolve the conjecture for the case n =
3 and all55
m ≥ 2; in other words, we prove that K̃m,3 has the capacity to
admit multiplenondegenerate steady states for all m ≥ 2 (Theorem
4.5). To accomplish this,we need information beyond the mere
existence of multiple steady states; we also
need precise values (or at least estimates) for the rates and
steady states. By
applying the proofs underlying the determinant optimization
method in Craciun60
and Feinberg’s work to the networks K̃m,n, we obtain (via
standard methods for
analyzing recurrence relations) explicit closed forms for
multistationary rates
and steady states. Then we use these closed forms to verify that
the steady
states are nondegenerate for small values of m and n.
Finally, recognizing the usefulness of generating closed forms
(or at least65
estimates1) for rates and steady states for any reaction network
that satisfies
the hypotheses of the determinant optimization method, in
Section 3 we outline
the steps of the method with enough generality to be used in
other contexts.
These steps are present in Craciun and Feinberg’s work but are
spread out
over several proofs, so our contribution here is to reorganize
the method into a70
concise procedure.
An outline of our work is as follows. Section 2 introduces
chemical systems
and the main conjecture. Section 3 describes the determinant
optimization
method in detail. We use this method to resolve some cases of
the main conjec-
ture in Sections 4 and 5. Finally, a discussion appears in
Section 6.75
Notation. We denote the positive real numbers by R+ := {x ∈ R |
x > 0},
1For general networks, the determinant optimization method need
not yield closed forms
for the rates and steady states, but one can nonetheless obtain
estimates.
3
-
and the standard inner product in Rn by 〈−,−〉. The ith entry of
a vector x isdenoted xi.
2. Background
This section introduces chemical reaction networks, their
corresponding mass-80
action kinetics systems, and the main object of our paper: the
sequestration
network K̃m,n.
2.1. Mass-action kinetics systems
Definition 2.1. A chemical reaction network G = {S, C,R}
consists of threefinite sets:85
1. a set of species S = {X1, X2, . . . , Xs},2. a set C of
complexes, which are non-negative integer linear combinations
of the species, and
3. a set R ⊆ C × C of reactions.
Example 2.2. The following chemical reaction network:
X1 + X2 → X3X2 → X1 + X4 ,
is entirely defined by:90
1. the set of species S = {X1, X2, X3, X4},2. the set of
complexes C = {X1 + X2, X3, X2, X1 + X4}, and3. the set of
reactions R = {(X1 + X2, X3), (X2, X1 + X4)}.
Any reaction network G = {S, C,R} is contained in the fully open
extensionnetwork G̃ obtained by including all inflow and outflow
reactions:95
G̃ :={S, C ∪ S ∪ {0} , R∪{Xi ↔ 0}Xi∈S
}. (1)
In other words, the fully open extension of any network is
obtained by adding
the reactions X → 0 (inflow) and 0→ X (outflow) for all X ∈ S.As
all the reactions take place, the concentrations of each of the
species
will change. We make use of mass-action kinetics to define a
system of ordi-
nary differential equations that describes, for each species,
how its concentration100
changes as a function of time. This ODE system is described by
the stoichio-
metric matrix Γ and the reactant vector R(x), which is a
vector-valued function
of the vector of species concentrations x.
4
-
Definition 2.3. Let G = {S, C,R} be a network, and let {y1 →
y′1, y2 →y′2, . . . , y|R| → y′|R|} be an ordering of the
reactions.105
1. The reaction vector of the reaction yi → y′i is the vector
y′i− yi, viewed inR|S|. Note that yi → y′i is a slight abuse of
notation, used to denote thereaction yi · X → y′i · X where X is
the vector of all species. Explicitly,the vectors yi and y
′i only contain species coefficients.
2. The stoichiometric matrix of G is the |S|×|R| matrix Γ whose
kth column110is the reaction vector of yk → y′k.
3. The reactant vector R(x) is the vector of length |R| whose
kth entry is the(monomial) product:
rkx(yk)11 x
(yk)22 · · ·x
(yk)|S||S| ,
where rk ∈ R+ is the reaction rate of the kth reaction.
Definition 2.4. The mass-action kinetics system of a network G =
{S, C,R}and a vector of reaction rates (rk) ∈ R|R|+ is defined by
the following system ofordinary differential equations:115
dx
dt= Γ ·R(x) . (2)
Example 2.5. For the following network:
A + 2Br→ 2A ,
Γ =
[1
−2
]and R(x) =
(rxAx
2B
), so the mass-action kinetics system (2) is:
[dxAdt
dxBdt
]= Γ ·R(x) =
[rxAx
2B
−2rxAx2B
].
An important characteristic of mass-action kinetics systems is
that they may
or may not have the capacity to admit (positive) steady
states:
Definition 2.6. A positive steady state is a vector x∗ ∈
R|S|>0 such that Γ ·R(x∗) = 0. A steady state x∗ is
nondegenerate if Im(df(x∗)|Im(Γ)) = Im(Γ),where df(x∗) denotes the
Jacobian matrix of the mass-action kinetics system at120
x∗.
5
-
Definition 2.7. A network that includes all inflow and outflow
reactions is mul-
tistationary2 if there exist two distinct concentration vectors
x∗,x# and positive
reaction rates such that Γ ·R(x∗) = Γ ·R(x#) = 0.
2.2. The sequestration network Km,n125
The main object of our paper is the fully open extension of the
network
Km,n:
Definition 2.8. For positive integers n ≥ 2 and m ≥ 2, the
sequestrationnetwork Km,n is:
X1 + X2 → 0 (3)
X2 + X3 → 0...
Xn−1 + Xn → 0
X1 → mXn .
K̃m,n is the fully open extension of Km,n, obtained by adjoining
all inflow and
outflow reactions, as in (1).
Schlosser and Feinberg analyzed variations of K̃2,n [11, Table
1], as did130
Craciun and Feinberg [6, Table 1.1]. Joshi and Shiu introduced
the version of
the sequestration networks in Definition 2.8, and proved that
some of them are
multistationary:
Proposition 2.9. [4, Lemma 6.9] For positive integers m ≥ 2 and
n ≥ 3, if nis odd, then K̃m,n admits multiple positive steady
states.135
For m = 1 or n even, the network K̃m,n is “injective” and
therefore not
multistationary [4, §6]. Joshi and Shiu conjectured that
Proposition 2.9 extendsas follows:
Conjecture 2.10. For positive integers m ≥ 2 and n ≥ 3, if n is
odd, thenK̃m,n admits multiple nondegenerate steady states.140
To resolve Conjecture 2.10, we must show that Im(df(x∗)) = Im(Γ)
for two
distinct positive steady states x∗. We will see in (4) below
that Γ is full rank,
so we need only show that det(df(x∗)) 6= 0 for two positive
steady states x∗.
2The focus of this work is on certain networks K̃mn, that
include all flow reactions. For net-
works that do not include all flow reactions, the definition of
multistationary must incorporate
the conservation relations in the network, if any.
6
-
Remark 2.11. As mentioned in the introduction, networks in which
all reac-
tants and products are at most bimolecular—that is, each complex
has the form145
0, X, X + Y , or 2X—are the norm in chemistry. This is the case
for the
networks K̃2,n, so that the nth internal reaction is X1 →
2Xn.
We end this section by displaying the matrices that define the
mass-action
kinetics system (2) defined by K̃m,n. We order the reactions as
follows: first, we
enumerate the n internal (or true) reactions listed in (3) (so,
the first reaction150
is X1 + X2 → 0, and so on), next are the n outflow reactions
(so, the (n + 1)streaction is X1 → 0, and so on), and then we have
the n inflow reactions (so, the(2n + 1)st reaction is 0 → X1, and
so on). We will refer to the sets of internal(true), outflow, and
inflow reactions as RT , RO, and RI , respectively.
The stoichiometric matrix for K̃m,n is:155
Γ =
−1 0 0 . . . 0 −1−1 −1 0 . . . 0 0
0 −1 −1. . .
...... −In In
0 0 −1. . .
......
......
. . .. . . −1 0
0 0 0 . . . −1 m
, (4)
where In is the n× n identity matrix. The reactant vector
is:
R(x) =
r1x1x2
r2x2x3...
rn−1xn−1xn
rnx1
rn+1x1
rn+2x2...
r2nxn
r2n+1...
r3n
,
where the ri ∈ R+ are the reaction rates and each xi ∈ R+ is the
concentration
7
-
of each species Xi. The mass-action ODEs (2.4) are:
ẋ1 =− r1x1x2 − rnx1 − rn+1x1 + r2n+1ẋi =− ri−1xi−1xi −
rixixi+1 − rn+ixi + r2n+i for 2 ≤ i ≤ n− 1
ẋn =− rn−1xn−1xn + mrnx1 − r2nxn + r3n .
Thus the Jacobian matrix, df(x), is the following (n×
n)-matrix
−r1x2 − rn − rn+1 −r1x1 0 . . . 0 0
−r1x2 −r1x1 − r2x3 − rn+2 −r2x2 . . ....
...
0 −r2x3 −r2x2 − r3x4 − rn+3. . . 0 0
... 0 −r3x4. . . −rn−2xn−2 0
0...
.... . . −rn−2xn−2 − rn−1xn − r2n−1 −rn−1xn−1
mrn 0 0 . . . −rn−1xn −rn−1xn−1 − r2n
.
(5)
3. Constructing multiple steady states via the determinant
optimiza-
tion method
The determinant optimization method3 was developed by Craciun
and Feinberg to160
show that certain chemical reaction networks are multistationary
[6]. More precisely,
the method guarantees that some networks (such as those that
satisfy the ‘Input’
conditions below) are necessarily multistationary. For instance,
Joshi and Shiu showed
that the networks K̃m,n (for m ≥ 2 and odd n ≥ 3) satisfy the
‘Input’ conditions, andthus concluded these networks are
multistationary (Proposition 2.9).165
In fact, the determinant optimization method also applies to
some networks that
do not satisfy the ‘Input’ conditions. To determine if this is
the case for a given
network, one must check whether a certain optimization problem
has a solution (see
Remark 3.4). If so, then the method guarantees that the network
is multistationary.
However, in many applications, it is useful not only to know
that a network is170
multistationary but also to have explicit steady-state
concentrations and reaction rates
that are witnesses to multistationarity. For instance, here we
would like to determine
whether the steady states are degenerate, whereas in other
settings one might like to
perform stability analysis.
Fortunately, the proofs in [6, §4] that underlie the determinant
optimization method175are constructive, up to one use of the
Intermediate Value Theorem, so one can generate
or at least approximate steady states and rates. This section
describes the step-by-
step procedure to do this; following our steps is easier than
(although equivalent to)
“backtracking” through the proofs. That is, our contribution
here is to re-package
the determinant optimization method into a constructive
algorithm. We will see that180
3Related techniques for establishing multistationarity appear in
work of Banaji and Pan-
tea [8, §4], Feliu [10, §2], and Müller et al. [9, §3.2].
8
-
for some networks, such as K̃m,n, the method constructs closed
forms for the steady
states and rates.
Determinant optimization method (constructive version)
Input: Any chemical reaction network G with n = |S| species that
contains all ninflow reactions such that there exist n reactions y1
→ y′1, y2 → y′2, . . . , yn → y′n185among the internal (true) and
outflow reactions RT ∪RO of G for which
(I) det(y1, y2, ..., yn) · det((y1 − y′1), (y2 − y′2), ..., (yn
− y′n)) < 0, and(II) there exists a vector η̃ ∈ Rn+ such that
Σni=1η̃i(yi − y′i) ∈ Rn+ = R
|S|+ .
Output: A certificate of multistationarity of G: (approximations
of) a positive re-
action rate vector (ry→y′) ∈ R|R|+ and two positive
concentration vectors x∗ and x#190which are both steady states of
the mass-action system defined by G and (ry→y′).
Steps: Described below.
With an eye toward resolving Conjecture 2.10, K̃m,n will be our
ongoing example.
Example 3.1. For K̃m,n (with m ≥ 2 and n ≥ 3 odd), hypothesis
(II) is satisfied bythe vector η̃ = (1, 1, . . . , 1,m+ 1, 1) [4,
Lemma 6.9]. Hypothesis (I) was proven in [4,195
Lemma 6.7], where the n reactions are precisely the n internal
(true) reactions (3).
Conveniently, these are the reactions labeled yi → y′i of the
sequestration network, for1 ≤ i ≤ n, so our notation for the first
n reactions—as well as the use of n for thenumber of
species—matches that of the determinant optimization method.
The steps below involve a certain linear transformation Tη;
specifically, for η ∈200RRT∪RO , the linear transformation Tη :
R|S| → R|S| is defined by:
Tη(δ) =∑
y→y′∈RT∪RO
ηy→y′(y · δ)(y − y′) . (6)
Equivalently, the matrix representation of Tη is d(−f)(1, 1, . .
. , 1) where the rates aregiven by ri = ηi. In other words, this
matrix is the Jacobian matrix of the mass-
action system (2.4) defined by the internal and outflow reaction
rates η (and any
choice of inflows: they do not appear in the Jacobian matrix) at
the concentration205
vector (1, 1, . . . , 1).
Example 3.2. For K̃m,n, the matrix representation of Tη is:
η1 + ηn + ηn+1 η1 0 · · · 0η1 η1 + η2 + ηn+2 η2 · · · 00 η2 η2 +
η3 + ηn+3 η3 0... 0
. . .. . .
...
0... · · · ηn−2 + ηn−1 + η2n−1 ηn−1
−mηn 0 · · · ηn−1 ηn−1 + η2n .
(7)
From the Jacobian matrix (5), it is clear that this matrix (7)
equals d(−f)(1, 1, . . . , 1),where the reaction rates are given by
ri = ηi.
9
-
The first step is to construct a (strictly positive) vector η− ∈
R|RT∪RO|+ , indexed by210all internal (true) and outflow reactions,
such that
(I) det(Tη−) < 0, and
(II)∑
y→y′∈RT∪ROη−y→y′(y − y
′) ∈ R|S|+ .
Craciun and Feinberg proved that these conditions (I) and (II)
are satisfied by a vector
η− of the following form:
η−y→y′ =
λη̃y→y′ if y → y′ ∈ {yi → y′i | i ∈ [n]}� else ,where λ is
sufficiently large and � is sufficiently small [6, proof of Theorem
4.2].
Example 3.3. For K̃m,n (with m ≥ 2 and n ≥ 3 odd), we define η−
as follows:
η−i =
λ if 1 ≤ i ≤ n− 2 or i = n
(m+ 1)λ if i = n− 1
� if n+ 1 ≤ i ≤ 2n .
Remark 3.4 (Stronger versions of the determinant optimization
method). This sec-215
tion describes the simplest version of the determinant
optimization method. In fact,
even if a network does not satisfy the hypotheses in the input
given above, the method
can still apply: [6, Remark 4.1] describes, in this setting, how
to implement the above
first step, i.e. how to test whether a suitable η− exists, as an
optimization problem.
Specifically, this is a polynomial optimization problem with
linear constraints over a220
compact set. Therefore, one can use any applicable optimization
method. Additionally,
see Remark 3.5 for how one can begin the algorithm at the second
step.
The second step is to construct a (strictly positive) vector η0
∈ R|RT∪RO|+ for which:
(I′) det(Tη0) = 0, and
(II′)∑
y→y′∈RT∪ROη0y→y′(y − y′) ∈ R
|S|+ .225
Craciun and Feinberg proved that this can be accomplished as
follows [6, proof of
Theorem 4.1]. First, construct an η+ ∈ R|RT∪RO|+ such that
det(Tη+) > 0; do this byassigning a large value to outflow
reactions and a small value to internal reactions:
η+y→y′ =
λ+ if y → y′ ∈ RO�+ if y → y′ ∈ RT ,where λ+ > 0 is large and
�+ > 0 is small.
Then, by interpolating between this vector η+ and the vector η−
from the pre-
vious step, the Intermediate Value Theorem (plus the fact that
the set of vectors
satisfying condition (II) is convex) guarantees the existence of
an η0 with the required
properties. Moreover, such a suitable η0 can be numerically
approximated, and, with230
careful tracking of error, one can use this approximation to
generate steady states and
concentrations in the following steps.
10
-
Remark 3.5 (Interpretation of the second step and subsequent
steps). What the
second step does is to find reaction rates (given by η0 for the
internal and outflow
rates, and the vector in (II ′) for the inflow rates) at which
the concentration vector235
(1, 1, . . . , 1) is a degenerate steady state: degeneracy is by
(I ′), and being a steady state
comes from (II ′).
Equivalently, if (r̃y→y′) is any positive vector of reaction
rates at which some con-
centration vector c̃ is a degenerate positive steady state, then
the vector η ∈ R|RT∪RO|+defined coordinate-wise by ηy→y′ = r̃y→y′
c̃
y satisfies the second step. So, a reader240
who has already found a degenerate positive steady state of
their system could start the
determinant optimization method at the second step. In other
words, one could begin
applying the method by immediately searching for a suitable η0
(without first generating
η− and η+). One strategy for doing this is described in Remark
3.6, which we employ
for K̃m,n beginning in Example 3.7.245
In the next steps, the determinant optimization method
constructs a certain vector
δ so that |δ| is a suitable bifurcation parameter: for |δ| small
but positive, the degeneratesteady state breaks into two
nondegenerate steady states.
Remark 3.6. Here is one strategy for constructing a suitable η0
(without using η−
and η+). First, identify (if possible) an η ∈ R|S|+ and a
reaction yi → y′i among the250internal (true) and outflow reactions
such that:
(a)∑
y→y′∈(RT∪RO)\{yi→y′i}η0y→y′(y − y′) ∈ R
|S|+ , and
(b) yi − y′i ∈ R|S|≥0 (this holds, for instance, if yi → y
′i is an outflow reaction).
One could see whether η+ or η− might work (we use η− in Example
3.7 below). Then,
define η as follows: let the entry η0i (corresponding to the
same ith reaction) be free, and255
fix η0j = η−j for j 6= i. Then, solve the (univariate
polynomial) equation det(Tη0) = 0.
If there is a positive solution (for η0i ), then the resulting
vector η0 is positive, and (I ′)
holds by construction. Furthermore, (II ′) holds because the sum
in (II ′) is precisely the
sum of a positive vector (namely, the sum in (a)) and a
non-negative vector (namely,
η0i (yi − y′i)). However, η0i is not guaranteed to be positive,
so this strategy may fail.260
Example 3.7. For K̃m,n (with m ≥ 2 and n = 3, 5, 7, 9, 11), the
following choice ofη0 satisfies the requirements of the second
step:
η0i =
λ if 1 ≤ i ≤ n− 2 or i = n
(m+ 1)λ if i = n− 1
� if n+ 1 ≤ i ≤ 2n−
1(m+1)(mλn+λ2(m+1)kn−2)(λ(m+2)+�)kn−2−λ2kn−3
− λ(m+ 1) if i = 2n ,
(8)
where λ > 0 and � > 0 are such that η02n is positive4
(thus, all coordinates of η0 are
4We checked that such λ, � exist for n = 3, 5, 7, 9, 11 (and m ≥
2), and we furthermoreconjecture that for larger n, choosing λ
sufficiently large and � sufficiently small will suffice.
11
-
positive), and ki is ith principal minor of the matrix
representation of Tη0 displayedbelow in (9), i.e. ki is the
determinant of the i× i (tridiagonal) upper-left submatrix265of (9)
(also, k0 := 1).
To show that this choice of η0 satisfies the two conditions of
the second step, we
first note that (II ′) is straightforward to verify.
Satisfying (I ′) only requires η0 to satisfy one (determinantal)
equation, so allowing
one free variable is sufficient. We choose η02n as this free
variable, and we will recover270
the formula in (8). Namely, we let all other coordinates η0i
have the form given above
(for 1 ≤ i ≤ 2n− 1), and then, recalling (7), the matrix
representation of Tη0 is:
2λ+ � λ 0 · · · 0λ 2λ+ � λ 0 · · · 0
0 λ. . .
. . ....
... 0. . . 0
0... λ+ λ(m+ 1) + � λ(m+ 1)
−mλ 0 · · · 0 λ(m+ 1) λ(m+ 1) + η02n
. (9)
Expanding (9) along the bottom row, we obtain the determinant of
Tη0 :
det(Tη0) = (−1)nm(m+ 1)λn − λ2(m+ 1)2kn−2 + (λ(m+ 1) + η02n)kn−1
, (10)
where we recall that ki is ith principal minor of Tη0 .The
determinant of tridiagonal matrices can be solved recursively [12].
For our
matrix, it is easy to verify the following recursion for i ≤ n−
2, which is independentof m:
ki+2 = (2λ+ �)ki+1 − λ2ki ,
with initial values k0 := 1 and k1 = 2λ+�. Notice that kn−1 must
be treated separately,because the (n − 1)st row contains η0n−1,
which is a function of m. Using standardmethods we can get the
generating function of the recurrence.
ki =1
2i+1c1∗(−c2(c2 − c1)i + c1(c2 − c1)i + c2(c1 + c2)i + c1(c1 +
c2)i
),
where c1 = (�)12 (� + 4λ)
12 and c2 = � + 2λ. Notice here that ki is always positive
for sufficiently small �. A formula for kn−1 is given using the
formula for tridiagonalmatrices [12]:
kn−1 = kn−2(λ(m+ 2) + �)− λ2kn−3 .
With this recurrence solved, we set equation (10) to zero and
then derive the explicit275
function for η02n from (10) in terms of m,λ, and �; this is the
formula in (8).
The third step is to construct a nonzero vector δ ∈ R|S| in the
nullspace of Tη0 , i.e.such that Tη0 · δ = 0. (Such a vector δ
exists because det(Tη0) = 0.)
12
-
Example 3.8. For our example K̃m,n (with m ≥ 2 and n ≥ 3 odd),
we claim that thevector δ whose coordinates are defined as follows
is in the nullspace of (9):280
δk =
δ1 if k = 1
−(2λ+�)λ
δk−1 − δk−2 if 2 ≤ k ≤ n− 1−(λ(m+2)+�)
λ(m+1)δn−1 − 1m+1δn−2 if k = n ,
(11)
where we introduce δ0 = 0 for convenience in solving the
recurrence relation, and
δ1 6= 0 is our free variable. Note that the last coordinate, δn,
has a different formula,because the (n− 1)st row in (9) used to
define δn contains terms dependent on m thatdo not satisfy the
recurrence.
To see that δ is a nonzero vector in the nullspace of Tη0 ,
notice that the conditions285
on δ that state that its inner product with each of the the
first n−1 rows of Tη0 coincideprecisely with the n− 1 recurrences
in the definition of δ (11). We claim that the lastrow of Tη is
linearly dependent on the other rows, and from this we will
conclude that
the last row of Tη0 automatically has zero inner product with δ.
To see this, we recall
that detTη0 = 0 by construction, so we need only show that the
first n− 1 rows of Tη0290are linearly independent. Assume, to the
contrary, that there is a non-trivial linear
combination of the first n − 1st rows of Tη0 that adds to 0.
Note that in the firstn− 1 rows only the last one contains an entry
in the last column, namely η0n−1. Sinceη0n−1 6= 0, the
corresponding scalar of the n−1st row should be 0, thus
annihilating theentire row. In the same manner, the corresponding
scalar for the n−2nd row would be295equal to 0. Repeating the
process shows that the only linear combination that adds to
0 is the trivial one, thus, arriving at a contradiction. So, δ
is in the nullspace of Tη0 .
Again, by using standard techniques for analyzing recurrences,
we find the gener-
ating function for each of the first n− 1 entries of δ:
δk = δ1λ ·(√
4λ�+ �2 − (2λ+ �))k − (−√
4λ�+ �2 − (2λ+ �))k
2kλk√
4λ�+ �2,
for 1 ≤ k ≤ n− 1.
The final step is to use the vectors η0 and δ (or, as we will
see, a sufficiently scaled
version of δ) from the previous two steps to construct a
certificate of multistation-
arity [6, proof of Lemma 4.1]; namely, the internal (true) and
outflow reaction rates
are:
ry→y′ =〈y, δ〉
e〈y,δ〉 − 1η0y→y′ for all y → y′ ∈ RT ∪RO ,
the inflow reaction rates are the coordinates of the following
vector:
(r0→Xi) =∑y→y′
η0y→y′(y − y′) ∈ R|S|+ , (12)
and the two steady states are:
x∗ = (1, 1, ..., 1) and x# = (eδ1 , eδ2 , ..., eδ|S|) .
13
-
Craciun and Feinberg showed that for sufficiently small scaling
of δ, all inflow rates (12)300
are positive [6].
Example 3.9. For K̃m,n, with m ≥ 2 and n = 3, 5, 7, 9, 11, we
checked that δ1 = 1suffices. Details for the n = 3 case are
provided in Remark 4.2.
Summarizing what we accomplished above, we have closed-form
expressions for
reaction rate constants and steady states that show that the
sequestration network is305
multistationary:
Theorem 3.10. Consider positive integers m ≥ 2 and n ∈ {3, 5, 9,
11}. Let δ ∈ Rn
be as in (11) with δ1 = 1. Also, let η0 be as is (8)5. Then, for
the following internal
(true) and outflow reaction rates:
ri =〈yi, δ〉
e〈yi,δ〉 − 1η0i for all i ∈ {1, 2, ..., 2n} ,
and the following inflow reaction rates:
r2n+1 = r1 + rn + rn+1
r2n+i = ri−1 + ri + rn+i for all 2 ≤ i ≤ n− 1
r3n = rn−1 + r2n −mrn ,
the concentrations:
x∗ = (1, 1, ..., 1) and x# = (eδ1 , eδ2 , ..., eδn) (13)
both are positive steady states of the mass-action kinetics
system defined by K̃m,n and
the reaction rates ri above.
Remark 3.11. One may wonder whether or not the determinant
optimization method
has the potential to create degenerate steady states. Indeed, if
we could prove that310
this method always constructs nondegenerate steady states, then
this would resolve
Conjecture 2.10. However, this is not the case.
We determined this by analyzing K̃2,3 as in Theorem 3.10. By
letting � be a free
variable we compute the parametrized determinants det(df(x∗))
and det(df(x#)) as
functions of �. Both functions are easily checked to be
continuous for positive values315
of �, and from the graph (Figure 1), we can see easily that
there exist choices of �
for which one of the two steady states is degenerate. More
precisely det(df(x∗)) = 0
for some � ∈ (0.12, 0.125) and det(df(x#)) = 0 for some � ∈
(0.240, 0.241) and some� ∈ (1.159, 1.160).
5In fact, this theorem will hold for any larger n for which the
last coordinate of η0 as
defined in (8) can be made to be positive.
14
-
Figure 1: Graphs of det(df(x∗)) solid and det(df(x#)) dashed as
functions of �
.
4. Resolving Conjecture 2.10 for the n = 3 case320
Recall that Conjecture 2.10 asserts that K̃m,n admits multiple
nondegenerate pos-
itive steady states, for integers m ≥ 2 and n ≥ 3 with n odd.
The main result of thissection (Theorem 4.5) resolves the
conjecture when n = 3. To accomplish this, we first
write down rate constants for this n = 3 case for which there
are two steady states
x∗ and x#; these values were obtained by the determinant
optimization method in325
the previous section for the general K̃m,n case (Proposition
4.1). We then resolve the
conjecture for n = 3 by proving that x∗ and x# are
nondegenerate.
4.1. Reaction rate constants for which K̃m,3 is
multistationary
Proposition 4.1 below specializes Theorem 3.10 to the n = 3
case. Following the
description in Section 3, λ = 1 and � = 0.1 will suffice, and
then we obtain
η0 =
(λ, λ(m+ 1), λ, �, �,
m2 − 0.31m− 1.312.1m+ 3.41
)Tfrom the second step of the determinant optimization method.
Next, in the third step,
we find that the following vector spans the nullspace of Tη0
:
δ =
(1, − 2.1, 2.1m+ 3.41
m+ 1
)T.
Thus, Theorem 3.10 specializes to:
15
-
Proposition 4.1. Consider any integer m ≥ 2, and the following
internal, outflow,330and internal reaction rates:
r1 =−1.1
e−1.1−1 ≈ 1.65 r2 =1.31
e1.31m+1−1
r3 =1e−1 ≈ .58
r4 =.1e−1 ≈ .06 r5 =
−.21e−2.1−1 ≈ .24 r6 =
m−1.31
e2.1m+3.41
m+1 −1r7 = r1 + r3 + r4 ≈ 2.29 r8 = r1 + r2 + r5 r9 = r2 + r6
−mr3 .
(14)
Then for the mass-action kinetics system defined by the fully
open sequestration net-
work K̃m,3 and the above rate constants ri, both x∗ = (1, 1, 1)
and x# =
(e, e−2.1, e
2.1m+3.41m+1
)are positive steady states.
Note that in Proposition 4.1, only x#3 , r2, r6, r8, and r9
depend on m.335
Remark 4.2. The only reaction rate in (14) that is not obviously
positive is the inflow
rate r9, so we verify it here:
r9 = r2 + r6 −mr3 > r2 + 0−m(
1
e− 1
)≥ m−m
(1
e− 1
)> 0 ,
where the second-to-last inequality follows from Lemma 4.3
below.
4.2. Bounding rates and steady states of K̃m,3
Here we give upper and lower bounds which we will use to prove
that x∗ and x#
are nondegenerate. The following bounds are on the third
coordinate of x#:
e2.1y+3.41
y+1 ≥ x#3 = e2.1m+3.41
m+1 > e2.1 for all m ≥ y ≥ 0 . (15)
The first inequality in (15) follows from the easy fact that
e2.1m+3.41
m+1 is a decreasing340
function when m > 0, and the second inequality is
straightforward.
The proofs of the following two upper/lower bounds are in
Appendix A:
Lemma 4.3 (Bounds on r2). When λ = 1 and � = 0.1, the rate
constant r2 defined
in (14) satisfies the following inequalities for all m ≥ 2:
m+ 1 > r2 ≥ m .
Lemma 4.4 (Bounds on r6). When λ = 1 and � = 0.1, the rate
constant r6 defined
in (14) satisfies the following inequalities:
0.14m > r6 > 0.13m− 0.5 ,
where the upper bound holds for m ≥ 2, and the lower bound holds
for m ≥ 20.
16
-
4.3. Proving nondegeneracy of steady states for the network
K̃m,3
The main result of this section is:345
Theorem 4.5 (Resolution of Conjecture 2.10 when n = 3). For
integers m ≥ 2, thenetwork K̃m,3 has the capacity to admit multiple
nondegenerate positive steady states.
We will prove Theorem 4.5 by showing that x∗ and x# in
Proposition 4.1 are
nondegenerate, i.e. we must prove that the image of the 3× 3
Jacobian matrix df(x)at each of the steady states is equal to the
image of the 3 × 9 matrix Γ. As stated350earlier (after Conjecture
2.10), since Γ is full rank, our problem reduces to showing
that det(df(x)) 6= 0 for both steady states and for all integers
m ≥ 2.We begin by displaying the Jacobian matrix (5) of K̃m,3:
df(x) =
−r1x2 − r3 − r4 −r1x1 0−r1x2 −r1x1 − r2x3 − r5 −r2x2mr3 −r2x3
−r2x2 − r6
.Thus, our goal is to show that the following determinants
(obtained by strategically
cancelling and rearranging terms) are nonzero for all integers m
≥ 2:
D1 := det(df(x∗)) = r2r1r3m − (r2 + r6)(r1r3 + r1r4 + r1r5 +
r3r5 + r4r5)
− r2r6(r1 + r3 + r4) (16)
D2 := det(df(x#)) = r2x
#2 ((r1x
#2 + r3 + r4)(r2x
#3 ) + r1x
#1 mr3)
− (r2x#2 + r6)(r1x#2 + r3 + r4)(r1x
#1 + r2x
#3 + r5)
+ (r2x#2 + r6)(r1x
#1 r1x
#2 ) (17)
From its graph6, D1 appears to be increasing quadratically as a
function of m. So,
to prove that D1 is nonzero for integer values of m ≥ 2, we will
bound it from below bya quadratic function. Similarly, D2 appears
to be decreasing quadratically, so we will355
bound it from above by another quadratic function. Using these
bounds, we will then
conclude that D1 and D2 are strictly positive and negative
(respectively) after certain
cutoff points of m, effectively showing nondegeneracy of both
steady states beyond the
cutoffs. Finally, we will complete the proof by evaluating D1
and D2 at the remaining
integers m between the 2 and the cutoff points to show that
these values are nonzero.360
Proof of Theorem 4.5. We generate our rates and concentrations
as in Proposition 4.1.
Following the description immediately after Theorem 4.5, we need
only show that
D1 6= 0 and D2 6= 0 for all integers m ≥ 2. First, we bound D1
by using its formula (16)together with the bounds in Lemmas 4.3 and
4.4:
D1 > m2r1r3 − ((m+ 1) + 0.14m)(r1r3 + r1r4 + r1r5 + r3r5 +
r4r5)
− (m+ 1)(0.14m)(r1 + r3 + r4) .
6Analogous graphs for larger n appear in Appendix B.
17
-
Next, estimating the remaining rates ri, which are constants
(recall equations (14)),
by appropriate upper or lower bounds, we obtain:
D1 > m2(0.95)− ((m+ 1) + 0.14m)1.61− (m+ 1)(0.14m)2.29
= 0.6294m2 − 2.156m− 1.61 .
It is easy to show that the quadratic function which bounds D1
above is always positive
for integers m > 4. So, D1 > 0 for m > 4. Thus, it
remains only to show that D1 6= 0at m = 2, 3, 4; indeed, those
values are nonzero and are listed in Table 1.
m 2 3 4 5 6 7 8 9 10
D1 = det(df(x∗)) 0.336 2.784 6.525
D2 = det(df(x#)) -1.063 -3.811 -7.85 -13.19 -19.8 -27.71 -36.89
-47.36 -59.11
m 11 12 13 14 15 16 17 18 19
D2 = det(df(x#)) -72.14 -86.4 -102 -118.9 -137.1 -156.5 -177.2
-199.2 -222.5
Table 1: Determinants of the Jacobian matrices (16–17) at the
two steady states x∗ and x#
for the values of m ≥ 2 before the proven bounds are valid. All
of these determinants arenonzero, so the corresponding steady
states are nondegenerate.
Now we proceed to bound D2. Again, we use its formula (17)
together with the365
bounds in (15) and Lemmas 4.3 and 4.4:
D2 < (m+ 1)x#2 ((r1x
#2 + r3 + r4)((m+ 1)x
#3 ) + r1x
#1 mr3)
− (mx2 + (.13m− .5))(r1x#2 + r3 + r4)(r1x#1 +mx
#3 + r5)
+ ((m+ 1)x#2 + .14m)(r1x#1 r1x
#2 ) .
Note that we used the lower bound on r6, so the above inequality
holds for m ≥ 20(and thus we will need to check the values of m
between 2 and 19 separately). In the
same manner as before, we approximate all of the constants
appropriately for m ≥ 20,and then simplify:
D2 < (m+ 1).13((.85)((m+ 1)8.7) + 2.61m)− (.25m−
.5)(.84)(4.72 + 8.16m)
+ (.13(m+ 1) + .14m)(.91)
= −0.41295m2 + 4.9437m+ 3.06205 .
Therefore, it is easy to see that D2 is nonzero for m ≥ 20. For
2 ≤ m ≤ 19 we againrefer to Table 1, which completes our proof.
5. Resolving Conjecture 2.10 for small values of m and n370
The main result of this section extends Theorem 4.5 to n ≤ 11,
for small m:
18
-
Theorem 5.1 (Resolution of Conjecture 2.10 for small m and n).
For m = 2, 3, 4, 5
and n = 5, 7, 9, 11, the network K̃m,n has the capacity to admit
multiple nondegenerate
positive steady states.
Proof. We generate our rates and concentrations as in Theorem
3.10: it is straightfor-375
ward to check that δ1 = 1, λ = 1, and � = 0.001 satisfy all
necessary hypotheses. Thus,
we obtain two steady states, x∗ = (1, 1, . . . , 1) and x#
defined in (13). Then, as in the
proof of Theorem 3.10, we verify that the determinants
det(df(x∗)) and det(df(x#))
are nonzero for m = 2, 3, 4, 5, which is readily seen from their
graphs, which appear in
Appendix B. (In fact, the graphs strongly suggest that the
conjecture holds completely380
for each of these values of n, namely n = 5, 7, 9, 11, i.e. for
m > 5 as well.)
6. Discussion
As stated in the introduction, deciding whether a chemical
reaction network is
multistationary is not easy in the general case. And even when
we can confirm that
a network is multistationary, there is no general technique to
show that it will admit385
multiple nondegenerate steady states. Nonetheless, in this paper
we succeeded in this
task for certain sequestration networks K̃m,n by using the
determininant optimization
method to obtain closed forms for reaction rates and steady
states.
Our work resolved the n = 3 case of Conjecture 2.10, and we
believe that our
results form an important step toward resolving the full
conjecture. Specifically, one390
could use the formulas for rates and steady states given in
Theorem 3.10 to analyze
the general case, or, perhaps easier, the case of some fixed m
and general n. Two
other possible approaches are to (1) find an alternate method to
obtain closed forms
for the steady states of a chemical reaction network, or (2)
identify criteria that can
guarantee that steady states are nondegenerate.395
Expanding on the last idea, our ultimate goal is to develop
general techniques
to assert that steady states of a chemical reaction network are
nondegenerate. For
instance, our analysis of the Jacobian determinants in this work
suggest that even
if the determinant optimization method yields a degenerate
steady state, then the
rate constants can be perturbed slightly so that the degenerate
steady state becomes400
nondegenerate (and the other steady state also remains
nondegenerate). Is this true
for any network for which the determinant optimization method
applies? If so, then
this would completely resolve Conjecture 2.10, and, more
generally, this would enable
us to more readily “lift” multistationarity and thereby enlarge
our catalogue of known
multistationary networks.405
Acknowledgements
BF and ZW conducted this research as part of the NSF-funded REU
in the Department
of Mathematics at Texas A&M University (DMS-1460766), in
which AS served as
mentor. All authors contributed substantially to this work. AS
was supported by
19
-
the NSF (DMS-1312473). The authors thank Dean Baskin for help
with the proof410
of Lemma 4.3, and Maya Johnson, Badal Joshi, Emma Owusu
Kwaakwah, Casian
Pantea, Xiaoxian Tang, and Jacob White for advice and fruitful
discussions. The
authors also thank an anonymous referee.
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Appendix A: proofs of Lemmas 4.3 and 4.4.
Lemma 6.1 (Lemma 4.3). The function r2(m) =1.31
e1.31m+1−1
satisfies the following
inequalities for all m ≥ 2:
m+ 1 > r2(m) ≥ m .
Proof. For the upper bound, first observe that
log
((1 +
1.31
m+ 1
)m+1)< log(e1.31) = 1.31 for all m ≥ 0 ,
since limx→∞(1 +yx
)x converges to ey from below for positive values of y.
Thus:
1.31 > log
((1 +
1.31
m+ 1
)m+1)
> log
((m+ 2.31
m+ 1
)m+1)= (m+ 1) log
(m+ 2.31
m+ 1
),
which implies that
1.31
m+ 1> log
(m+ 2.31
m+ 1
)=⇒ e
1.31m+1 >
m+ 2.31
m+ 1
=⇒ e1.31m+1 (m+ 1)− (m+ 1) > 1.31 .
and this final inequality implies our desired upper bound: m+1
> 1.31
e1.31m+1−1
= r2(m).
Notice that our lower bound is equivalent to the following
inequality:
m+ 1.31
m≥ e
1.31m+1 for all m ≥ 2 . (18)
21
http://www.sciencedirect.com/science/article/B6TFK-44CRH1D-5C/2/4368d27af5ad95131107f482727432a8http://www.sciencedirect.com/science/article/B6TFK-44CRH1D-5C/2/4368d27af5ad95131107f482727432a8http://www.sciencedirect.com/science/article/B6TFK-44CRH1D-5C/2/4368d27af5ad95131107f482727432a8http://dx.doi.org/DOI:
10.1016/0009-2509(94)80061-8http://www.sciencedirect.com/science/article/B6TFK-44CRH1D-5C/2/4368d27af5ad95131107f482727432a8http://www.sciencedirect.com/science/article/B6TFK-44CRH1D-5C/2/4368d27af5ad95131107f482727432a8http://www.sciencedirect.com/science/article/B6TFK-44CRH1D-5C/2/4368d27af5ad95131107f482727432a8http://scholarworks.rit.edu/cgi/viewcontent.cgi?article=2122&context=articlehttp://scholarworks.rit.edu/cgi/viewcontent.cgi?article=2122&context=articlehttp://scholarworks.rit.edu/cgi/viewcontent.cgi?article=2122&context=articlehttp://scholarworks.rit.edu/cgi/viewcontent.cgi?article=2122&context=articlehttp://scholarworks.rit.edu/cgi/viewcontent.cgi?article=2122&context=articlehttp://scholarworks.rit.edu/cgi/viewcontent.cgi?article=2122&context=articlehttp://www.desmos.com
-
We set a = 1.31, make use of the change of variables z = 1m
, and then apply log to
see that our desired inequality (18) is equivalent to:
log(1 + az) ≥ az−1 + 1
=az
1 + zfor all z ∈ (0, 1/2) .
We will show that log(1 + az)− az1+z≥ 0. To this end, define b
by 1− b = a− 1, and
notice that 1 > b > 12> (a− 1). Next, note that we have
the following equalities:
log(1 + az) −az
1 + z=
∫ az0
(1
1 + t−
1
1 + z
)dt
=
∫ bz0
z − t
(1 + z)(1 + t)dt +
∫ zbz
(1
1 + t−
1
1 + z
)+
∫ azz
z − t
(1 + z)(1 + t)dt . (19)
The second integral in (19) is nonnegative (because its
integrand is nonnegative), so
we complete the proof now by showing that the sum of the first
and third integrals
in (19) is nonnegative:∫ bz0
z − t(1 + z)(1 + t)
dt+
∫ azz
z − t(1 + z)(1 + t)
dt ≥ (1− b)z2b
(z + 1)2+−(a− 1)2z2
(z + 1)2≥ 0 ,
where the two inequalities come from recalling that b < 1,
and, respectively, (1− b) =455(a− 1) and b ≥ (a− 1).
Lemma 6.2 (Lemma 4.4). The function r6(m) =m−1.31
e2.1m+3.41
m+1 −1satisfies:
0.14m > r6(m) > 0.13m− 0.5 , (20)
where the upper bound holds for m ≥ 2, and the lower bound holds
for m ≥ 20.
Proof. We first prove the upper bound. By the second inequality
in (15), (e2.1m+3.41
m+1 −1) > 0 for m ≥ 2. Thus our desired upper bound in (20)
is equivalent to the following:
m(0.14e2.1m+3.41
m+1 − 1.14) > − 1.31 ,
which holds (for positive m) whenever (0.14e2.1m+3.41
m+1 − 1.14) > 0. This inequality inturn is equivalent to the
following (since log is an increasing function):
2.1m+ 3.41 > (m+ 1) log
(1.14
0.14
)≈ (m+ 1)(2.10) ,
which is true for positive m, so the proof of the upper bound is
complete.
For the lower bound, by clearing the denominator and gathering
exponential terms
on the right-hand side, we see that the desired inequality is
equivalent to the following:
1.13m− 1.81 ≥ e2.1m+3.41
m+1 (0.13m− 0.5) .
We prove this now. The first inequality below is equivalent to
the inequality .00004m ≥−2.536, which is true for positive m:
1.13m− 1.81 ≥ 8.692(0.13m− 0.5)
> e2.1(20)+3.41
(20)+1 (0.13m− 0.5) ≥ e2.1m+3.41
m+1 (0.13m− 0.5) ,
and the final inequality holds for m ≥ 20 because e2.1m+3.41
m+1 is a decreasing function
for positive values of m.460
22
-
Appendix B: graphs for the proof of Thoeorem 5.1
Figures 2a–3a below present the graphs of the determinant of the
Jacobian matrix
evaluated at the steady states x∗ and x# (as described in the
proof of Theorem 5.1)
for the network K̃m,n for odd 5 ≤ n ≤ 11 as functions of m. Note
that the graphsare nonzero for 2 ≤ m ≤ 5, confirming Conjecture
2.10 for odd 5 ≤ n ≤ 11 and those465values of m. Also, the graphs
strongly suggest that the conjecture holds for larger m
as well. All graphs were made using Desmos Graphing Calculator
[13].
(a) det(df(x∗)) (b) det(df(x#))
Figure 2: K̃m,5
(a) det(df(x∗)) (b) det(df(x#))
Figure 3: K̃m,7
(a) det(df(x∗)) (b) det(df(x#))
Figure 4: K̃m,9
23
-
(a) det(df(x∗)) (b) det(df(x#))
Figure 5: K̃m,11
24
IntroductionBackgroundMass-action kinetics systemsThe
sequestration network Km,n
Constructing multiple steady states via the determinant
optimization methodResolving Conjecture 2.10 for the n=3
caseReaction rate constants for which K"0365Km,3 is
multistationaryBounding rates and steady states of
K"0365Km,3Proving nondegeneracy of steady states for the network
K"0365Km,3
Resolving Conjecture 2.10 for small values of m and
nDiscussion
